In my opinion, Paul is making the same point in the verse cited in the question as he did in Romans 5:13, when he said, "Sin indeed was in the world before the Law was given, but sin is not counted where there is no law."
Prior to the giving of the Law, sin existed as a fact (that is, people were sinful and died as a consequence of Adam's original sin), but they were not directly disobeying a command from God, as Adam had done.
That changed when God gave the Law through Moses, since, after that, people knew what specific actions were forbidden, and what the penalties were for disobedience.
To me, this is shown by individuals prior to the giving of the Law who took actions that the Law later said were forbidden, but whom God did not punish for their actions, because the Law had not yet been given.
For example, Abraham, whom God selected as the father of His chosen people, had married his own half-sister (Sarah)(Genesis 20:12), which the Law later prohibited (Leviticus 18:9). Also, God did not punish Jacob for marrying two sisters (Leah and Rachel) (Genesis 29:15-30), which the Law later similarly forbade (Leviticus 18:18).

In my humble opinion, according to scripture, the law (God's Ten Commandments) had been in existence long before Sinai. (Read Genesis (9:5,6: 18:19; 26:5; 39:7-10) Israel kept the Sabbath before Sinai (Exodus. 16:22-26)
What happened on Mt. Sinai was to redirect the Israelites back to God and the grace He offers all who come to Him by faith.
God did not need to reveal His law to Abraham (Genesis 26:5) with thunder, lightning, and a penalty of death (Exodus 19:10-23). God did give the law to the Israelites in this manner. It was because, during their bondage in Egypt, the Israelites had lost sight of God's greatness and His high moral standards. Because of this, they needed to be made aware of the extent of their own sinfulness and sacredness of God's law. The revelation at Sinai did just that.

God's Moral Law, the Ten Commandments (TC) existed as long as sin has existed. The Bible says, "Where no law is, there is no transgression" (Rom 4:15). So God's TC law existed from the beginning. Men broke that law (sinned, 1 John 3:4) and because of sin (or breaking God's law), Moses' law was given (or "added" Gal 3:16, 19) till Christ should come and die.
Between Genesis 1 and Exodus 20, there is a period of approximately 2500 years. It is difficult to imagine that all this time, God did not teach His creation His Moral Law - the Ten Commandments, and people went on killing, lying, fornicating or stealing without any consequence. But we do not have to imagine or speculate because the Bible make several references to the TC long before Exodus 20, when the Law was given.
Cain could have not been guilty of murder if God did not say “Thou shalt not kill” (Ex 20:13; Gen 4:7).
Joseph didn’t have to be concerned with “Thou shalt not commit adultery” with Potiphar’s wife if there was no law about adultery (Gen 39:9; Ex 20:14).
Furthermore, Abraham kept God’s commandments (Gen 26:5) long before Exodus 20.
Moses' law, or the ceremonial law of the Old Testament, was given to deal with the priesthood, sacrifices, circumcision, rituals, meat and drink offerings, etc., all of which foreshadowed the cross. This law was added "till the seed should come," and that seed was Christ (Gal 3:16, 19). The ritual and ceremony of Moses' law pointed forward to Christ's sacrifice. When Christ died, this law came to an end, but the Ten Commandments (God's law) "stand fast for ever and ever." Ps 111:8. That there are two laws is made crystal clear in Daniel 9:10-11.
In the Book of Galatians, Paul settles the issue of circumcision – meaning that the rite of circumcision is not required for salvation, since in Christ all are justified by faith. Here Paul Is not addressing the TC law – he is addressing exclusively circumcision which was part of the ceremonial law, totally separate issue from the TC (Gal. 5:6; Gal. 6:15 Gal. 6:12-13).